8+e=0.2e+0.8e(e-10)

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Solution for 8+e=0.2e+0.8e(e-10) equation:


Simplifying
8 + e = 0.2e + 0.8e(e + -10)

Reorder the terms:
8 + e = 0.2e + 0.8e(-10 + e)
8 + e = 0.2e + (-10 * 0.8e + e * 0.8e)
8 + e = 0.2e + (-8e + 0.8e2)

Combine like terms: 0.2e + -8e = -7.8e
8 + e = -7.8e + 0.8e2

Solving
8 + e = -7.8e + 0.8e2

Solving for variable 'e'.

Combine like terms: e + 7.8e = 8.8e
8 + 8.8e + -0.8e2 = -7.8e + 0.8e2 + 7.8e + -0.8e2

Reorder the terms:
8 + 8.8e + -0.8e2 = -7.8e + 7.8e + 0.8e2 + -0.8e2

Combine like terms: -7.8e + 7.8e = 0.0
8 + 8.8e + -0.8e2 = 0.0 + 0.8e2 + -0.8e2
8 + 8.8e + -0.8e2 = 0.8e2 + -0.8e2

Combine like terms: 0.8e2 + -0.8e2 = 0.0
8 + 8.8e + -0.8e2 = 0.0

Begin completing the square.  Divide all terms by
-0.8 the coefficient of the squared term: 

Divide each side by '-0.8'.
-10 + -11e + e2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -11e + 10 + e2 = 0 + 10

Reorder the terms:
-10 + 10 + -11e + e2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -11e + e2 = 0 + 10
-11e + e2 = 0 + 10

Combine like terms: 0 + 10 = 10
-11e + e2 = 10

The e term is -11e.  Take half its coefficient (-5.5).
Square it (30.25) and add it to both sides.

Add '30.25' to each side of the equation.
-11e + 30.25 + e2 = 10 + 30.25

Reorder the terms:
30.25 + -11e + e2 = 10 + 30.25

Combine like terms: 10 + 30.25 = 40.25
30.25 + -11e + e2 = 40.25

Factor a perfect square on the left side:
(e + -5.5)(e + -5.5) = 40.25

Calculate the square root of the right side: 6.34428877

Break this problem into two subproblems by setting 
(e + -5.5) equal to 6.34428877 and -6.34428877.

Subproblem 1

e + -5.5 = 6.34428877 Simplifying e + -5.5 = 6.34428877 Reorder the terms: -5.5 + e = 6.34428877 Solving -5.5 + e = 6.34428877 Solving for variable 'e'. Move all terms containing e to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + e = 6.34428877 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + e = 6.34428877 + 5.5 e = 6.34428877 + 5.5 Combine like terms: 6.34428877 + 5.5 = 11.84428877 e = 11.84428877 Simplifying e = 11.84428877

Subproblem 2

e + -5.5 = -6.34428877 Simplifying e + -5.5 = -6.34428877 Reorder the terms: -5.5 + e = -6.34428877 Solving -5.5 + e = -6.34428877 Solving for variable 'e'. Move all terms containing e to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + e = -6.34428877 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + e = -6.34428877 + 5.5 e = -6.34428877 + 5.5 Combine like terms: -6.34428877 + 5.5 = -0.84428877 e = -0.84428877 Simplifying e = -0.84428877

Solution

The solution to the problem is based on the solutions from the subproblems. e = {11.84428877, -0.84428877}

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